To see all the transactions (i.e. payments) into and out of your wallet, select "Payments" from the sidebar.
New transactions (payments or receipts) are first broadcast to the Bitcoin network, and are later mined into blocks.
For a new payment, MultiBit HD sends the transaction to some of the peers it is connected to and then listens to the other peers for the transaction being retransmitted. So the most basic status that MultiBit HD shows about a transaction is the fact that it's been successfully broadcast to the Bitcoin network.
The next step is for the transaction to be confirmed, i.e. included in a block that becomes a new link in the block chain. But since the block chain is an evolving, decentralised database, there is no central authority that says what is or is not the end of the chain. It is a bit like when you want to join the end of a queue, but you're not sure who is at the end.
MultiBit HD describes a transaction as confirmed when it is included in one mined block, i.e. something that is a valid new end of the block chain. The more deeply buried that block is in the block chain (i.e. the more blocks that queue up after it and refer to it), the more certain you can be that the transaction won't disappear, won't be reversed, and is in fact a permanent part of the block chain.
MultiBit HD suggests a level of certainty you can have about a transaction by showing as the transactions's "Status" a pie chart building up the number of blocks in the block chain that come after and refer to the transaction. It is up to you to decide how many blocks you need to be certain that a transaction is permanent, but MultiBit HD suggests 6 blocks, and when there are 6 blocks, MultiBit HD shows a tick against the status. You can always see the precise number of blocks by double-clicking on the transaction.